ON THE 2 LARGEST EIGENVALUES OF TREES

Very little is known about upper bounds for the largest eigenvalues of a tree that depend only on the vertex number. Starting from a classic al upper bound for the largest eigenvalue, some refinements can be obt ained by successively removing trees from consideration. The results c an be used to characterize those trees that maximize the second larges t eigenvalue. This corrects a result from the literature, and it inclu des a proof of a conjecture of Neumaier. The main tool for this endeav or is the theory of partial engenvectors. (C) Elsevier Science Inc., 1 997 .